On mean values of some zeta-functions in the critical strip
Journal de Théorie des Nombres de Bordeaux, Tome 15 (2003) no. 1, pp. 163-178.

Un entier $k\ge 3$ et un réel $\sigma$ tel que $\frac{1}{2}<\sigma <1$ étant fixés, on considère dans la formule asymptotique

 ${\int }_{1}^{T}{\left|\zeta \left(\sigma +it\right)\right|}^{2k}dt=\sum _{n=1}^{\infty }{d}_{k}^{2}\left(n\right){n}^{-2\sigma }T+R\left(k,\sigma ;T\right),$
le terme erreur $R\left(k,\sigma ;T\right)$, pour lequel nous montrons de nouvelles bornes lorsque min $\left({\beta }_{k},{\sigma }_{k}^{*}\right)<\sigma <1$. Nous obtenons également des majorations nouvelles pour les termes erreur dans le développement des moments d’ordre deux des fonctions zêta de formes paraboliques holomorphes et des séries de Rankin-Selberg.

For a fixed integer $k\ge 3$, and fixed $\frac{1}{2}<\sigma <1$ we consider

 ${\int }_{1}^{T}{\left|\zeta \left(\sigma +it\right)\right|}^{2k}dt=\sum _{n=1}^{\infty }{d}_{k}^{2}\left(n\right){n}^{-2\sigma }T+R\left(k,\sigma ;T\right),$
where $R\left(k,\sigma ;T\right)=0\left(T\right)\left(T\to \infty \right)$ is the error term in the above asymptotic formula. Hitherto the sharpest bounds for $R\left(k,\sigma ;T\right)$ are derived in the range min $\left({\beta }_{k},{\sigma }_{k}^{*}\right)<\sigma <1$. We also obtain new mean value results for the zeta-function of holomorphic cusp forms and the Rankin-Selberg series.

@article{JTNB_2003__15_1_163_0,
author = {Ivi\'c, Aleksandar},
title = {On mean values of some zeta-functions in the critical strip},
journal = {Journal de Th\'eorie des Nombres de Bordeaux},
pages = {163--178},
publisher = {Universit\'e Bordeaux I},
volume = {15},
number = {1},
year = {2003},
zbl = {1050.11075},
mrnumber = {2019009},
language = {en},
url = {http://www.numdam.org/item/JTNB_2003__15_1_163_0/}
}
Ivić, Aleksandar. On mean values of some zeta-functions in the critical strip. Journal de Théorie des Nombres de Bordeaux, Tome 15 (2003) no. 1, pp. 163-178. http://www.numdam.org/item/JTNB_2003__15_1_163_0/

 R. Balasubramanian, A. Ivić, K. Ramachandra, An application of the Hooley-Huxley contour. Acta Arith. 65 (1993), 45-51. | MR 1239242 | Zbl 0781.11035

 K. Chandrasekharan, R. Narasimhan, Functional equations with multiple gamma factors and the average order of arithmetical functions. Ann. of Math. 76 (1962), 93-136. | MR 140491 | Zbl 0211.37901

 K. Chandrasekharan, R. Narasimhan, The approximate functional equation for a class of zeta-functions. Math. Ann. 152 (1963), 30-64. | MR 153643 | Zbl 0116.27001

 A. Ivić, The Riemann zeta-function, John Wiley & Sons, New York, 1985. | MR 792089 | Zbl 0556.10026

 A. Ivić, Large values of certain number-theoretic error terms. Acta Arith. 56 (1990), 135-159. | MR 1075641 | Zbl 0659.10053

 A. Ivić, On zeta-functions associated with Fourier coefficients of cusp forms. Proceedings of the Amalfi Conference on Analytic Number Theory (Amalfi, September 1989), Università di Salerno, Salerno 1992, pp. 231-246. | MR 1220467 | Zbl 0787.11035

 A. Ivić, Some problems on mean values of the Riemann zeta-function. J. Théor. Nombres Bordeaux 8 (1996), 101-123. | Numdam | MR 1399949 | Zbl 0858.11045

 A. Ivić, On some conjectures and results for the Riemann zeta-function and Hecke series. Acta Arith. 99 (2001), no. 2, 115-145. | MR 1847617 | Zbl 0984.11043

 A. Ivić, M. Ouellet, Some new estimates in the Dirichlet divisor problem. Acta Arith. 52 (1989), 241-253. | MR 1031337 | Zbl 0619.10041

 A. Ivić, K. Matsumoto, On the error term in the mean square formula for the Riemann zeta-function in the critical strip. Monatsh. Math. 121 (1996), 213-229. | MR 1383532 | Zbl 0843.11039

 A. Ivić, K. Matsumoto, Y. Tanigawa, On Riesz means of the coefficients of the Rankin-Selberg series. Math. Proc. Camb. Phil. Soc. 127 (1999), 117-131. | MR 1692491 | Zbl 0958.11065

 S. Kanemitsu, A. Sankaranarayanan, Y. Tanigawa, A mean value theorm for Dirichlet series and a general divisor problem, to appear.

 K. Matsumoto, The mean values and the universality of Rankin-Selberg L-functions. Proc. Turku Symposium on Number Theory in Memory of K. Inkeri, May 31-June 4, 1999, Walter de Gruyter, Berlin etc., 2001, in print. | MR 1822011 | Zbl 0972.11075

 A. Perelli, General L-functions. Ann. Mat. Pura Appl. 130 (1982), 287-306. | MR 663975 | Zbl 0485.10030

 R.A. Rankin, Contributions to the theory of Ramanujan's function τ(n) and similar arithmetical functions. II, The order of Fourier coefficients of integral modular forms. Math. Proc. Cambridge Phil. Soc. 35 (1939), 357-372. | Zbl 0021.39202

 R.A. Rankin, Modular Forms. Ellis Horwood Ltd., Chichester, England, 1984. | MR 803359 | Zbl 0546.00010

 H.-E. Richert, Über Dirichletreihen mit Funktionalgleichung. Publs. Inst. Math. (Belgrade) 11 (1957), 73-124. | MR 92816 | Zbl 0082.05802

 A. Selberg, Old and new conjectures and results about a class of Dirichlet series. Proc. Amalfi Conf. Analytic Number Theory, eds. E. Bombieri et al., Università di Salerno, Salerno, 1992, pp. 367-385. | MR 1220477 | Zbl 0787.11037

 E.C. Titchmarsh, The theory of the Riemann zeta-function, 2nd edition.Oxford University Press, Oxford, 1986. | MR 882550 | Zbl 0601.10026

 Zhang Wenpeng, On the divisor problem. Kexue Tongbao 33 (1988), 1484-1485. | MR 969977